A concise course in complex analysis and Riemann surfaces. Wilhelm Schlag

Transcription

1 A concise course in complex analysis and Riemann surfaces Wilhelm Schlag

2

3 Contents Preface 5 Chapter 1. From i to z: the basics of complex analysis 9 1. The field of complex numbers 9 2. Differentiability and conformality Möbius transforms Integration Harmonic functions The winding number 31 Chapter 2. From z to the Riemann mapping theorem: some finer points of basic complex analysis The winding number version of Cauchy s theorem Isolated singularities and residues Analytic continuation Convergence and normal families The Mittag-Leffler and Weierstrass theorems The Riemann mapping theorem 5 7. Runge s theorem 54 Chapter 3. Harmonic functions on D The Poisson kernel Hardy classes of harmonic functions 6 3. Almost everywhere convergence to the boundary data 62 Chapter 4. Riemann surfaces: definitions, examples, basic properties The definition and examples Functions on Riemann surfaces 7 3. Degree and genus Riemann surfaces as quotients Elliptic functions 76 Chapter 5. Analytic continuation, covering surfaces, and algebraic functions Analytic continuation The unramified Riemann surface of an analytic germ The ramified Riemann surface of an analytic germ Algebraic germs and functions 92 3

4 4 CONTENTS Chapter 6. Differential forms on Riemann surfaces Holomorphic and meromorphic differentials Integrating differentials and residues The Hodge operator and harmonic differentials Statement and examples of the Hodge decomposition 113 Chapter 7. Hodge s theorem and the L 2 existence theory Weyl s lemma and the Hodge decomposition Existence of non-constant meromorphic functions 122 Chapter 8. The Riemann-Roch theorem Homology bases, periods, and Riemann s bilinear relations Divisors The proof of the Riemann-Roch theorem Applications and more general divisors 136 Chapter 9. The Dirichlet problem and Green functions Green functions The potential theory proof of the Riemann mapping theorem Existence of Green functions via Perron s method Behavior at the boundary 145 Chapter 1. Green functions and the classification problem Green functions on Riemann surfaces Hyperbolic Riemann surfaces admit Green functions 15 Chapter 11. The uniformization theorem The statement for simply connected surfaces Hyperbolic, simply connected, surfaces Parabolic, simply connected, surfaces 158 Chapter 12. Problems 163 Chapter 13. Hints and Solutions 181 Chapter 14. Review of some facts from algebra and geometry Geometry and topology Algebra 212 Bibliography 213

5 Preface During their first year at the University of Chicago, graduate students in mathematics take classes in algebra, analysis, and geometry, one of each every quarter. The analysis classes typically cover real analysis and measure theory, functional analysis, and complex analysis. This book grew out of the author s notes for the complex analysis class which he taught during the Spring quarter of 27 and 28. The course covered elementary aspects of complex analysis such as the Cauchy integral theorem, the residue theorem, Laurent series, and the Riemann mapping theorem with Riemann surface theory. Needless to say, all of these topics have been covered in excellent textbooks as well as classic treatise. This book does not try to compete with the works of the old masters such as Ahlfors [1], Hurwitz Courant [19], Titchmarsh [37], Ahlfors Sario [2], Nevanlinna [32], Weyl [39]. Rather, it is intended as a fairly detailed yet fast paced guide through those parts of the theory of one complex variable that seem most useful in other parts of mathematics. There is no question that complex analysis is a corner stone of the analysis education at every university and each area of mathematics requires at least some knowledge of it. However, many mathematicians never take more than an introductory class in complex variables that often appears awkward and slightly outmoded. Often, this is due to the omission of Riemann surfaces and the assumption of a computational, rather than geometric point of view. Therefore, the authors has tried to emphasize the very intuitive geometric underpinnings of elementary complex analysis which naturally lead to Riemann surface theory. As for the latter, today it is either not taught at all or sometimes given a very algebraic slant which does not appeal to more analytically minded students. This book intends to develop the subject of Riemann surfaces as a natural continuation of the elementary theory without which the latter would indeed seem artificial and antiquated. At the same time, we do not overly emphasize the algebraic aspect such as elliptic curves. The author feels that those students who wish to pursue this direction will be able to do so quite easily after mastering the material in this book. Because of such omissions as well as the reasonably short length of the book it is to be considered intermediate. Partly because of the fact that the Chicago first year curriculum covers topology and geometry this book assumes knowledge of basic notions such as homotopy, the fundamental group, differential forms, co-homology and homology, and from algebra we require knowledge of the notions of groups 5

6 6 PREFACE and fields, and some familiarity with the resultant of two polynomials (but the latter is needed only for the definition of the Riemann surfaces of an algebraic germ). However, only the most basic knowledge of these concepts is assumed and we collect the few facts that we do need in Chapter 14. Let us now describe the contents of the individual chapters in more detail. Chapter 1 introduces the concept of differentiability over C, the calculus of z, z, Möbius (or fractional linear) transformations and some applications of these transformations to hyperbolic geometry. In particular, we prove the Gauss-Bonnet theorem in that case. Next, we develop integration and Cauchy s theorem in various guises, then apply this to the study of analyticity, and harmonicity, the logarithm and the winding number. We conclude the chapter with some brief comments about co-homology and the fundamental group. Chapter 2 refines the Cauchy formula by extending it to zero homologous cycles, i.e., those cycles which do not wind around any point outside of the domain of holomorphy. We then classify isolated singularities, prove the Laurent expansion and the residue theorems with applications. After that, Chapter 2 studies analytic continuation and presents the monodromy theorem. Then, we turn to convergence of analytic functions and normal families with application to the Mittag-Leffler and Weierstrass theorems in the entire plane, as well as the Riemann mapping theorem. The chapter concludes with Runge s theorem. In Chapter 3 we study the Dirichlet problem on the unit disk. This means that we solve the boundary value problem for the Laplacian on the disk via the Poisson kernel. We present the usual L p based Hardy classes of harmonic functions on the disk, and discuss the question of representing them via their boundary data both in the L p and the almost every sense. We then sketch the more subtle theory of homolomorphic functions in the Hardy class, or equivalently of the boundedness properties of the conjugate harmonic functions (with the F.& M. Riesz theorem and the notion of inner and outer functions being the most relevant here). The theory of Riemann surfaces begins with Chapter 4. This chapter covers the basic definitions of such surfaces and the analytic functions on them. Elementary results such as the Riemann-Hurwitz formula for the branch points are discussed and several examples of surfaces and analytic functions defined on them are presented. In particular, we show how to define Riemann surfaces via discontinuous group actions and give examples of this procedure. The chapter closes with a discussion of tori and some aspects of the classical theory of meromorphic functions on these tori (doubly periodic or elliptic functions). Chapter 5 presents another way in which Riemann surfaces arise naturally, namely via analytic continuation. Historically, the desire to resolve unnatural issues related to multi-valued functions (most importantly for algebraic functions) lead Riemann to introduce his surfaces. Even though the underlying ideas leading from a so-called analytic germ to its Riemann

7 PREFACE 7 surface are very geometric and intuitive (and closely related to covering spaces in topology), their rigorous rendition requires some patience as ideas such as analytic germ, branch point, (un)ramified Riemann surface of an analytic germ etc., need to be defined precisely. This typically proceeds via some factorization procedure of a larger object (i.e., equivalence classes of sets which are indistinguishable from the point of view of the particular object we wish to construct). The chapter also develops some basic aspects of algebraic functions and their Riemann surfaces. At this point the reader will need to be familiar with the resultant of two polynomials. In particular, we will see that every (!) compact Riemann surface is obtained through analytic continuation of some algebraic germ. This uses the machinery of Chapter 5 together with a potential theoretic result that guarantees the existence of a non-constant meromorphic function on every Riemann surface, which we prove in Chapter 7. Chapter 6 introduces differential forms on Riemann surfaces and their integrals. Needless to say, the only really important class are the 1-forms and we define harmonic, holomorphic and meromorphic forms and the residues in the latter case. Furthermore, the Hodge operator appears naturally. We then present some examples that lead up to the Hodge decomposition in the next chapter. This refers to the fact that every 1-form can be decomposed additively into three components: a closed, co-closed, and a harmonic form (the latter being characterized as being simultaneously closed and co-closed). In this book, we follow the classical L 2 -based derivation of this theorem. Thus, via Hilbert space methods one first derives this decomposition with L 2 -valued forms and then uses Weyl s regularity lemma (weakly harmonic functions are smoothly harmonic) to upgrade to smooth forms. The proof of the Hodge theorem is presented in Chapter 7. This chapter includes a theorem on the existence of meromorphic differentials and functions on a general Riemann surface. In particular, we derive the striking fact that every Riemann surface carries a non-constant meromorphic function which is needed to complete the result on compact surfaces being algebraic in Chapter 5. Chapter 8 presents the well-known Riemann-Roch theorem which computes the dimension of certain spaces of meromorphic differentials from properties of the so-called divisor and the genus of the underlying compact Riemann surface. Before proving the theorem, there are a number of prerequisites to be dealt with, such as the Riemann period relations and the definition of a divisor. The remaining Chapters 9, 1 and 11 are devoted to the proof of the uniformization theorem. This theorem states that the only simply connected Riemann surfaces (up to isomorphisms) are C, D, and CP 1. For the compact case, we deduce this from the Riemann-Roch theorem. But for the other two cases we use methods of potential theory which are motivated by the proof of the Riemann mapping theorem which is based on the existence of a Green function. It turns out that such a function only exists for the hyperbolic

8 8 PREFACE surfaces (such as D) but not for the parabolic case (such as C) or the compact case. Via the Perron method, we prove the existence of a Green function for hyperbolic surfaces, thus establishing the conformal equivalence with the disk. For the parabolic case, a suitable substitute for the Green function needs to be found. We discuss this in detail for the simply connected case, and also sketch some aspects of the non-simply connected cases.

9 CHAPTER 1 From i to z: the basics of complex analysis 1. The field of complex numbers The field C of complex numbers is obtained by adjoining i to the field R or reals. The defining property of i is i = and complex numbers z 1 = x 1 + iy 1 and z 2 = x 2 + iy 2 are added component wise and multiplied according to the rule z 1 z 2 = x 1 x 2 y 1 y 2 + i(x 1 y 2 + x 2 y 1 ) which follows from i = and the distributional law. The conjugate of z = x + iy is z = x iy and we have z 2 := z z = x 2 + y 2. Therefore every z has a multiplicative inverse given by 1 z := z z 2 and C becomes a field. Since complex numbers z can be represented as points or vectors in R 2 in the Cartesian way, we can also assign polar coordinates (r, θ) to them. By definition, z = r and z = r(cos θ + i sin θ). The addition theorems for cosine and sine imply that z 1 z 2 = z 1 z 2 (cos(θ 1 + θ 2 ) + i sin(θ 1 + θ 2 )) which reveals the remarkable fact that complex numbers are multiplied by multiplying their lengths and adding their angles. In particular, z 1 z 2 = z 1 z 2. This shows that power series behave as in the real case with respect to convergence, i.e., a n z n converges on z < R and diverges for every z > R n= R 1 = lim sup a n 1 n n where the sense of convergence is relative to the length metric on complex numbers which is the same as the Euclidean distance on R 2 (the reader should verify the triangle inequality); the formula for R of course follows from comparison with the geometric series. Note that the convergence is absolute on the disk z < R and uniform on every compact subset of that disk. Moreover, the series diverges for every z > R as can be seen by the a comparison test. We can also write R = lim n n a n+1 provided this limit exists. The first example that comes to mind here is 1 1 z = z n, z < 1 n= 9

10 1 1. BASIC COMPLEX ANALYSIS I Another example is of course (1.1) E(z) := which converges absolutely and uniformly on every compact subset of C. Expanding (z 1 + z 2 ) n via the binomial theorem shows that E(z 1 + z 2 ) = E(z 1 )E(z 2 ). Recall the definition of the Euler constant e: consider the ordinary differential equation (ODE) ẏ = y with y() = 1 which has a unique solution y(t) for all t R. Then set e := y(1). Let us solve our ODE iteratively (Picard method). Thus, y(t) = 1 + = n j= t t j j! + n= y(s) ds = 1 + t + t z n n! (t s) n y(s) ds t (t s)y(s) ds =... The integral on the right vanishes as n and we obtain t j y(t) = j! j= which in particular yields the usual series expansion for e. group property of flows, y(t 2 )y(t 1 ) = y(t 1 + t 2 ) Also, by the which proves that y(t) = e t for every rational t and motivates why we define e t t j := t R j! j= Hence, our series E(z) above is used as definition of e z for all z C. We have the homomorphism property e z 1+z 2 = e z 1 e z 2, and by comparison with the power series of cos and sin on R, we arrive at the famous Euler formula e iθ = cos(θ) + i sin(θ) for all θ R. This in particular shows that z = re iθ where (r, θ) are the polar coordinates of z. This in turn implies that (cos θ + i sin θ) n = cos(nθ) + i sin(nθ) for every n 1 (de Moivre s formula). Now suppose that z = re iθ with r >. Then by the preceding, z = e log r+iθ or log z = log r + iθ Note that the logarithm is not well defined since θ and θ+2πn for any n Z both have the property that exponentiating leads to z. Similarly, ( 1 r n e i θ 2πi ) n n e n = z

11 2. DIFFERENTIABILITY AND CONFORMALITY 11 which shows that there are n different possibilities for n z. Later on we shall see how these functions become single valued on their natural Riemann surfaces. Let is merely mention at this point that the complex exponential is most naturally viewed as the covering map { C C := C \ {} z e z where C is the universal cover of C. But for now, we of course wish to differentiate functions defined on some open set Ω C. There are two relevant notions of derivative here and we will need to understand how they relate to each other. 2. Differentiability and conformality The first is the crucial linearization idea from multivariable calculus and the second copies the idea of difference quotients from calculus. In what follows we shall either use U or Ω to denote planar regions, i.e., open and connected subsets of R 2. Also, we will identify z = x + iy with the real pair (x, y) and will typically write a complex valued function as f = u + iv = (u, v). Definition 1.1. (a) We say that f C 1 (Ω) iff there exists df C(Ω), a matrix valued function, such that f(x + h) = f(x) + df(x)(h) + o(h) h (b) We say that f is holomorphic on Ω if f (z) := lim w z f(w) f(z) w z exists for all z Ω and is continuous on Ω. We denote this by f H(Ω). A function f H(C) is called entire. Note that (b) is equivalent to the existence of a function f C(Ω) so that f(z + h) = f(z) + f (z)h + o(h) h where f (z)h is the product between the complex numbers f (z) and h. Hence, we conclude that the holomorphic functions are precisely those C 1 (Ω) functions in the sense of (a) for which the differential df(x) acts as linear map via multiplication by a complex number. Obvious examples of holomorphic maps are the powers f(z) = z n for all n Z (if n is negative, then we exclude z = ). They satisfy f (z) = nz n 1 by the binomial theorem. Also, since we can do algebra in C the same way we did over R it follows that the basic differentiation rules like sum, product, quotient, and chain rules continue to hold for holomorphic functions. Let us demonstrate this for the chain rule: if f H(Ω), g H(Ω ) and f : Ω Ω, then we know from the C 1 chain rule that (f g)(z + h) = (f g)(z) + Df(g(z))Dg(z)h + o(h) h

12 12 1. BASIC COMPLEX ANALYSIS I From (b) above we infer that Df(g(z)) and Dg(z) act as multiplication by the complex numbers f (g(z)) and g (z), respectively. Thus, we see that f g H(Ω) and (f g) = f (g)g. We leave the product and quotient rules to the reader. It is clear that all polynomials are holomorphic functions. In fact we can generalize this to all power series within their disk of convergence. Let us make this more precise. Definition 1.2. We say that f : Ω C is analytic (or f A(Ω)) if f is represented by a convergent power series expansion locally around every point of Ω. Lemma 1.3. A(Ω) H(Ω) Proof. Suppose z Ω and f(z) = a n (z z ) n z z < r(z ) n= where r(z ) >. As in real calculus, one checks that differentiation can be interchanged with summation and in fact, f (z) = na n (z z ) n 1 z z < r(z ) n= In fact, we can of course differentiate any number of times and f (k) = (n) k a n (z z ) n k z z < r(z ) n= where (n) k = n(n 1)... (n k + 1). This proves also that a n = f (n) (z ) n! for all n. We note that with e z defined as above, (e z ) = e z from the series representation (1.1). It is a remarkable fact of basic complex analysis that A(Ω) = H(Ω). To establish equality here, we need to be able to integrate, see the section of integration below. Recall that f = u+iv = (u, v) belongs to C 1 (Ω) iff the partials u x, u y, v x, v y exist and are continuous on Ω. If f H(Ω), then (by letting w approach z along the x or y directions, respectively) f (z) = u x + iv x = iu y + v y so that u x = v y and u y = v x which is known as the Cauchy Riemann equations. They are equivalent to the property that [ ] ux u df = y = ρa, ρ, A SO(2, R) v x v y In other words, at each point where a holomorphic function f has a nonvanishing derivative, its differential df is a conformal matrix: it preserves angles and the orientation between vectors. Conversely, if f C 1 (Ω) has

13 2. DIFFERENTIABILITY AND CONFORMALITY 13 the property that df is proportional to a rotation everywhere on Ω, then f H(Ω). Let us summarize these observations. Theorem 1.4. A complex valued function f C 1 (Ω) is holomorphic iff the Cauchy-Riemann (CR) system holds in Ω. This is equivalent to df being the composition of a rotation and a dilation (possibly by zero) at every point in Ω. Proof. We already saw the (CR) is necessary. Conversely, since f C 1 (Ω), we can write u(x + ξ, y + η) = u(x, y) + u x (x, y)ξ + u y (x, y)η + o( (ξ, η) ) v(x + ξ, y + η) = v(x, y) + v x (x, y)ξ + v y (x, y)η + o( (ξ, η) ) Using that u x = v y and u y = v x we obtain, with ζ = ξ + iη f(z + ζ) f(z) = (u x + iv x )(z)(ξ + iη) + o( ζ ) which of course proves that f (z) = u x + iv x = v y iu y as desired. The second part was already covered above. The following notion is of central importance here: Definition 1.5. A function f C 1 (Ω) is called conformal if and only if df in Ω and df preserves the angle and orientation at each point. Thus, the holomorphic functions are precisely those C 1 functions which are conformal at all points at which df. Note that f(z) = z is C 1 (C) but not holomorphic (since it reverses orientations). Also note that f(z) = z 2 doubles angles at z = (in the sense that curves crossing at at angle α get mapped onto curves intersecting at at angle 2α), so conformality is lost there. A particularly convenient and insightful way of distinguishing holomorphic functions from C 1 functions is given by the z, z calculus. Assume that f C 1 (Ω). Then the real-linear map df(z) can be written as the sum of a complex-linear and a complex anti-linear transformation (meaning that T (zv) = zt (v)), see Lemma 6.2 below. In other words, there exist complex numbers ζ 1 (x), ζ 2 (x) such that df(x) = ζ 1 (x) dz + ζ 2 (x) d z where dz is simply the identity map and d z the reflection about the real axis. We used here that all complex linear transformations on R 2 are given by multiplication by a complex number, whereas the complex anti-linear ones become complex linear by composing them with a reflection. To find ζ 1 and ζ 2 simply observe that 1 df(x) = f x dx + f y dy = f x 2 (dz + d z) + f 1 y (dz d z) 2i = 1 2 (f x if y ) dz (f x + if y ) d z =: z f dz + z f d z

14 14 1. BASIC COMPLEX ANALYSIS I In other words, f H(Ω) iff f C 1 (Ω) and z f = in Ω. One can immediately check that z f = is the same as the Cauchy-Riemann system. As an application of this formalism we record the following crucial fact: for any f H(Ω), d(f(z) dz) = z f dz dz + z f d z z = which means that f(z) dz is a closed differential form. This property is equivalent to the homotopy invariance of the Cauchy integral that we will encounter below. We leave it to the reader the verify the chain rules z (g f) = ( w g) f z f + ( w g) f z f z (g f) = ( w g) f z f + ( w g) f z f as well as the representation of the Laplacean = 4 2 z z. These ideas are of particular importance once we discuss differential forms on Riemann surfaces. To continue our introductory chapter, we next turn to the simple but important idea of extending the notion of analyticity to functions that take the value. In a similar vein, we can make sense of functions being analytic at z =. To start with, we define the one-point compactification of C, which we denote by C, with the usual basis of the topology; the neighborhoods of are the complements of all compact sets. It is intuitively clear that C S 2 in the homeomorphic sense. Somewhat deeper as well as much more relevant for complex analysis is the fact that C S 2 \ {p} as conformal equivalence where p S 2 is arbitrary. This is done via the wellknown stereographic projection, see the homework and Chapter 4 below as well as Figure 1.1. If the circle in that figure is the unit circle, N = (, 1), and X = (x, y), then Z = x 1 y as the reader will easily verify using similarity of triangles. This identifies the stereographic projection as the map Φ : S 2 \ {(,, 1)} C, X = (x 1, x 2, x 3 ) x 1 + ix 2 1 x 3 The stereographic projection preserves angles as well as circles, see Problem 4 in Section 12. We will see in Chapter 4 that C S 2 CP 1 in the sense of conformal equivalences, and each of these Riemann surfaces are called the Riemann sphere. Without going into details about the exact definition of a Riemann surface, we mention in passing that C is covered by two charts, namely (C, z) and (C \ {}, z 1 ), which both are homeomorphisms onto C. On the overlap region C the change of charts is given by the map z z 1 which is of course a conformal equivalence. It is now clear how to define holomorphic maps (1.2) f : C C, f : C C, f : C C First, we need to require that f is continuous in each case. This is needed in order to ensure that we can localize f to charts. Second, we require f to

15 3. MÖBIUS TRANSFORMS 15 N X O Z Figure 1.1. Stereographic projection be holomorphic relative to the respective charts. For example, if f(z ) = for some z C, then we say that f is holomorphic close to z if and only if 1 f(z) is holomorphic around z. To make sense of f being analytic at z = with values in C, we simply require that f( 1 z ) is holomorphic around z =. For the final example in (1.2), if f( ) =, then f is analytic around z = if and only if 1/f(1/z) is analytic around z =. We shall see later in this chapter that the holomorphic maps f : C C are constants (indeed, such a map would have to be entire and bounded and therefore constant by Liouville s theorem, see Corollary 1.19 below). On the other hand, the maps f : C C are precisely the meromorphic ones which we shall encounter in the next chapter. Finally, the holomorphic maps f : C C are precisely the rational functions P (z) Q(z) where P, Q are polynomials. To see this1, one simply argues that any such f is necessarily meromorphic with only finitely many poles in C and a pole at z =. 3. Möbius transforms If we now accept that the holomorphic, and thus conformal, maps C C are precisely the rational ones, it is clear how to identify the conformal automorphisms (or automorphisms) amongst these maps. Indeed, in that case necessarily P and Q both have to be linear which immediately leads to the following definition. Based on the argument of the previous paragraph 1 The reader should not be alarmed in case he or she does not follow these arguments they will become clear once this chapter and the next one has been read.

16 16 1. BASIC COMPLEX ANALYSIS I (which the reader for now can ignore if desired), the lemma identifies all automorphisms of C. Lemma 1.6. Every A GL(2, C) defines a transformation T A (z) := az + b [ ] a b cz + d, A = c d which is holomorphic as a map from C C. It is called a fractional linear or Möbius transformation. The map A T A only depends on the equivalence class of A under the relation A B iff A = λb, λ C. In other words, the family of all Möbius transformations is the same as (1.3) P SL(2, C) := SL(2, C)/{±Id} We have T A T B = T A B and T 1 A = T A 1. In particular, every Möbius transform is an automorphism of C. Proof. It is clear that each T A is a holomorphic map C C. The composition law T A T B = T A B and T 1 A = T A 1 are simple computations that we leave to the reader. In particular, T A has a conformal inverse and is thus an automorphism of C. If T A = Tà where A, à SL(2, C), then T A(z) = ad bc (cz + d) 2 = T à (z) = ã d b c ( cz + d) 2 and thus cz + d = ±( cz + d) under the assumption that ad bc = ã d b c = 1 Hence, A and à are the same matrices in SL(2, C) possibly up to a choice of sign, which establishes (1.3). Fractional linear transformations enjoy many important properties which can be checked separately for each of the following four elementary transformations. In particular, Lemma 1.7 proves that the group P SL(2, C) has four generators. Lemma 1.7. Every Möbius transformation is the composition of four elementary maps: translations z z + z dilations z λz, λ > rotations z e iθ z, θ R inversion z 1 z Proof. If c =, then T A (z) = a d z + b d. If c, then and we are done. T A (z) = bc ad c 2 1 z + d c + a c

17 3. MÖBIUS TRANSFORMS 17 The reader will have no difficulty verifying that z z 1 z+1 take the right half-plane on the disk D := { z < 1}. In particular, ir gets mapped on the unit circle. Similarly, z 2z 1 2 z takes D onto itself with the boundary going onto the boundary. If we include all lines into the family of circles (they are circles passing through ) then these examples can serve to motivate the following lemma. Lemma 1.8. Fractional linear transformations take circles onto circles. Proof. In view of the previous lemma, the only case requiring an argument is the inversion. Thus, let z z = r be a circle and set w = 1 z. Then = z 2 2Re ( zz ) + z 2 r 2 = 1 w 2 2Re (wz ) w 2 + z 2 r 2 If z = r, then one obtains the equation of a line in w. Note that this is precisely the case when the circle passes through the origin. Otherwise, we obtain the equation = w z 2 r 2 z 2 r 2 ( z 2 r 2 ) 2 which is a circle. A line is given by an equation 2Re (z z ) = a which transforms into 2Re (z w) = a w 2. If a =, then we simply obtain another line through the origin. Otherwise, we obtain the equation w z /a 2 = z /a 2 which is a circle. An alternative argument uses the fact that stereographic projection preserves circles, see homework problem #4. To use it, note that the inversion z 1 z corresponds to a rotation of the Riemann sphere about the x 1 axis (the real axis of the plane). Since such a rotation preserves circles, a fractional linear transformation does, too. Since T z = z is a quadratic equation for any Möbius transform T, we see that T can have at most two fixed points unless it is the identity. It is also clear that every Möbius transform has at least one fixed point. The map T z = z +1 has exactly one fixed point, namely z =, whereas T z = 1 z has two, z = ±1. Lemma 1.9. A fractional linear transformation is determined completely by its action on three distinct points. Moreover, given z 1, z 2, z 3 C distinct, there exists a unique fractional linear transformation T with T z 1 =, T z 2 = 1, T z 3 =. Proof. For the first statement, suppose that S, T are Möbius transformations that agree at three distinct points. Then S 1 T has three fixed

18 18 1. BASIC COMPLEX ANALYSIS I points and is thus the identity. For the second statement, let T z := z z 1 z 2 z 3 z z 3 z 2 z 1 in case z 1, z 2, z 3 C. If any one of these points is, then we obtain the correct formula by passing to the limit here. Definition 1.1. The cross ratio of four points z, z 1, z 2, z 3 C is defined as [z : z 1 : z 2 : z 3 ] := z z 1 z 2 z 3 z z 3 z 2 z 1 This concept is most relevant for its relation to Möbius transformations. Lemma The cross ratio of any four distinct points is preserved under Möbius transformations. Moreover, four distinct points lie on a circle iff their cross ratio is real. Proof. Let z 1, z 2, z 3 be distinct and let T z j = w j for T a Möbius transformation and 1 j 3. Then for all z C, [w : w 1 : w 2 : w 3 ] = [z : z 1 : z 2 : z 3 ] provided w = T z This follows from the fact that the cross ratio on the left-hand side defines a Möbius transformation S 1 w with the property that S 1 w 1 =, S 1 w 2 = 1, S 1 w 3 =, whereas the right-hand side defines a transformation S with S z 1 =, S z 2 = 1, S z 3 =. Hence S 1 1 S = T as claimed. The second statement is an immediate consequence of the first and the fact that if any three distinct points z 1, z 2, z 3 R, then a fourth point z has a real-valued cross ratio with these three iff z R. We can now define what it means for two points to be symmetric relative to a circle (or line recall that this is included in the former). Definition Let z 1, z 2, z 3 Γ where Γ C is a circle. We say that z and z are symmetric relative to Γ iff [z : z 1 : z 2 : z 3 ] = [z : z 1 : z 2 : z 3 ] Obviously, if Γ = R, then z = z. In other words, if Γ is a line, then z is the reflection of z across that line. If Γ is a circle of finite radius, then we can reduce matters to this case by an inversion. Lemma Let Γ = { z z = r}. Then for any z C, z = r2 z z Proof. It suffices to consider the unit circle. Then [z; z 1 ; z 2 : z 3 ] = [ z : z 1 1 : z 1 2 : z 1 3 ] = [1/ z : z 1 : z 2 : z 3 ] In other words, z = 1 z. The general case now follows from this via a translation and dilation.

19 3. MÖBIUS TRANSFORMS 19 Figure 1.2. Geodesics in the hyperbolic plane Möbius transformations are important for several reasons. We already observed that they are precisely the automorphisms of the Riemann sphere (but to see that every automorphism is a Möbius transformation requires material from this entire chapter as well as the next). In the 19th century there was much excitement surrounding non-euclidean geometry and there is an important connection between Möbius transforms and hyperbolic geometry: the isometries of the hyperbolic plane H are precisely those Möbius transforms which preserve it. Let us be more precise. Consider the upper half-plane model of the hyperbolic plane given by H = {z C : Im z > }, ds 2 = dx2 + dy 2 y 2 = d zdz (Im z) 2 It is not hard to see that the subgroup of P SL(2, C) which preserves the upper half-plane is precisely P SL(2, R). Indeed, z az+b cz+d preserves R := C R if and only if a, b, c, d R. In other words, the stabilizer of R (as a set) is P GL(2, R) which contains P SL(2, R) as an index two subgroup. The latter preserves the upper half plane, whereas those matrices with negative determinant interchange the upper with the lower half-plane. It is easy to check (see the home work problems) that P SL(2, R) operates transitively on H and preserves the metric: for the latter, one simply computes w = az + b d w dw = cz + d (Im w) 2 = d z dz (Im z) 2 In particular, the geodesics are preserved under P SL(2, R). Since the metric does not depend on x it follows that all vertical lines are geodesics. By the transitive action of the group we conclude that all geodesics are generated

20 2 1. BASIC COMPLEX ANALYSIS I from these by applying group elements. Therefore, the geodesics of H are precisely all circles which intersect the real line at a right angle (with the vertical lines being counted as circles of infinite radius). From this it is clear that the hyperbolic plane satisfies all axioms of Euclidean geometry with the exception of the parallel axiom: there are many lines (i.e., geodesics) passing through a point which is not on a fixed geodesic that do not intersect that geodesic. Let us now prove the famous Gauss-Bonnet theorem which describes the hyperbolic area of a triangle whose three sides are geodesics (those are called geodesic triangles). This is of course a special case of a much more general statement about integrating the Gaussian curvature over a geodesic triangle on a general surface. The reader should prove the analogous statement for spherical triangles. Figure 1.3. Geodesic triangles Theorem Let T be a geodesic triangle with angles α 1, α 2, α 3. Then Area(T ) = π 3 j=1 α j. Proof. There are four essentially distinct types of geodesic triangles, depending on how many of its vertices lie on the real axis. Up to equivalences via transformations in P SL(2, R) (which are isometries and therefore also preserve the area) we see that it suffices to consider precisely those cases described in Figure 1.3. Let us start with the case in which exactly one vertex is in R as shown in that figure (the second triangle from the left). Without loss of generality this vertex coincides with 1 and the circular arc lies on the unit circle with the projection of the second finite vertex onto

21 the real axis being at x. Then Area(T ) = = 4. INTEGRATION 21 1 x α dxdy 1 dx y(x) y 2 = x 1 x 2 d cos φ 1 cos 2 (φ) = α = π α 1 as desired since the other two angles are zero. By additivity of the area we can deal with the other two cases in which at least one vertex is real. We leave the case where no vertex lies on the (extended) real axis to the reader, the idea is to use Figure 1.4. A B C D Figure 1.4. The case of no real vertex We leave it to the reader to generalize the Gauss-Bonnet theorem to geodesic polygons. Many interesting questions about Möbius transformations remain, for example how to characterize those that correspond to rotations of the sphere, or how to determine all finite subgroups of P SL(2, C). For some answers see Chapter 12 as well as [21]. A whole topic onto itself are the Fuchsian and Kleinian groups, see for example [24]. These groups are of crucial importance for the uniformization theory of Riemann surfaces in the non-simply connected case. 4. Integration We now develop our complex calculus further. The following definition defines the complex integral canonically in the sense that it is the only definition which preserves the fundamental theorem of calculus for holomorphic functions. Definition For any C 1 -curve γ : [, 1] Ω and any compexvalued f C(Ω) we define 1 f(z) dz = f(γ(t))γ (t) dt γ If γ is a closed curve (γ() = γ(1)) then we also write γ f(z) dz for this integral.

22 22 1. BASIC COMPLEX ANALYSIS I From the chain rule, we deduce the fundamental fact that the line integrals of this definition do not depend on any particular C 1 parametrization of the curve as long as the orientation is preserved (hence, there is no loss in assuming that γ is parametrized by t 1). Again from the chain rule, we immediately obtain the following: if f H(Ω), then f (z) dz = f(γ(1)) f(γ()) γ for any γ as in the definition. In particular, f (z) dz = closed curves γ in Ω γ On the other hand, let us compute with γ r (t) := re it, r >, 2π { (1.4) z n dz = r n e int rie it n 1 dt = 2πi n = 1 γ r In Ω = C, the function f(z) = z n has the primitive F n (z) = zn+1 n+1 provided n 1. This explains why we obtain for all n 1. On the other hand, if n = 1 we realize from our calculation that 1 z does not have a (holomorphic) primitive in C. This issue merits further investigation (for example, we need to answer the question whether 1 z has a local primitive in C this is indeed the case and this primitive is a branch of log z). Before doing so, however, we record the famous Cauchy theorem in its homotopy version. Figure 1.5 γ 2 γ 1 γ 3 Figure 1.5. Homotopy

23 4. INTEGRATION 23 shows two curves, namely γ 1 and γ 2, which are homotopic relative to the annular region they lie in. The dashed curve is not homotopic to either of them within the annulus. Theorem Let γ, γ 1 : [, 1] Ω be C 1 curves 2 with γ () = γ 1 () and γ (1) = γ 1 (1) (the fixed endpoint case) or γ j () = γ j (1), j =, 1 (the closed case). Assume that they are homotopic in the following sense: there exists a continuous map H : [, 1] 2 Ω with H(t, ) = γ (t), H(t, 1) = γ 1 (t) and such that H(, s) is a C 1 curve for each s 1. Moreover, in the fixed endpoint case we assume that H freezes the endpoints, whereas in the closed case we assume that each curve from the homotopy is closed. Then f(z) dz = f(z) dz γ for all f H(Ω). In particular, if γ is a closed curve in Ω which is homotopic to a point, then f(z) dz = γ Proof. We first note the important fact that f(z) dz is a closed form. Indeed, d(f(z) dz) = z f(z) dz dz + z f(z) d z dz = by the Cauchy-Riemann equation z f =. Thus, Cauchy s theorem is a special case of the homotopy invariance of the integral over closed forms which in turn follows from Stokes s theorem. Let us briefly recall the details: since a closed form is locally exact, we first note that f(z) dz = η for all closed curves η which fall into sufficiently small disks, say. But then we can triangulate the homotopy so that f(z) dz f(z) dz = f(z) dz = γ 1 j η j γ where the sum is over a finite collection of small loops which constitute the triangulation of the homotopy H. The more classically minded reader may prefer to use Green s formula (which of course follows from the Stokes theorem): provided U Ω is a sufficiently small neighborhood which is γ 1 2 This can be relaxed to piece-wise C 1 which means that we can write the curve as a finite union of C 1 curves. The same comment applies to the homotopy.

24 24 1. BASIC COMPLEX ANALYSIS I diffeomorphic to a disk, say, one can write f(z) dz = u dx v dy + i(u dy + v dx) U U = U ( u y v x ) dxdy + i ( v y + u x ) dxdy = U where the final equality sign follows from the Cauchy Riemann equations. This theorem is typically applied to very simple configurations, such as two circles which are homotopic to each other in the region of holomorphy of some function f. As an example, we now derive the following fundamental fact of complex analysis which is intimately tied up with the n = 1 case of (1.4). Proposition Let D(z, r) Ω and f H(Ω). Then (1.5) f(z) = 1 2πi γ for all z D(z, r). f(ζ) ζ z dζ where γ(t) = z + re it Proof. Fix any z D(z, r) and apply Theorem 1.16 to the region U ε := D(z, r) \ D(z, ε) where ε > is small. We use here that the two boundary circles of U ε are homotopic to each other relative to the region Ω. Then = 1 2πi = 1 2πi U ε 1 2πi f(ζ) z ζ dζ = 1 2πi D(z,r) D(z,r) D(z,ε) f(ζ) f(z) z ζ f(ζ) z ζ dζ dζ f(z) 2πi D(z,ε) f(ζ) dζ + O(ε) f(z) as ε z ζ 1 z ζ dζ where we used the n = 1 case of (1.4) to pass to the last line. We can now derive the astonishing fact that holomorphic functions are in fact analytic. This is done by noting that the integrand in (1.5) is analytic relative to z. Corollary A(Ω) = H(Ω). In fact, every f H(Ω) is represented by a convergent power series on D(z, r) where r = dist(z, Ω).

25 4. INTEGRATION 25 Proof. We already observe that analytic functions are holomorphic. For the converse, we use the previous proposition to conclude that f(z) = 1 f(ζ) 2πi ζ z (z z ) dζ = 1 2πi = n= γ γ 1 2πi f(ζ) ζ z γ n= ( z z ζ z ) n dζ f(ζ) (ζ z ) n+1 dζ (z z ) n where the interchange of summation and integration is justified due to uniform and absolute convergence of the series. Thus, we obtain that f is analytic and, moreover, f (n) f(z) = n! (z z ) n n= converges on z z < dist(z, Ω) with (1.6) f (n) (z ) = n! f(ζ) dζ 2πi (ζ z ) n+1 for any n. γ In contrast to power series over R, over C there is an explanation for the radius of convergence: f(z) = n= a n (z z ) n has finite and positive radius of convergence R iff f H(Ω) for every Ω which compactly contains D(z, R). We immediately obtain a number of corollaries from this. Corollary (a) Cauchy s estimates: Let f H(Ω) with f(z) M on Ω. Then f (n) Mn! (z) dist(z, Ω) n for every n and all z Ω. (b) Liouville s theorem: If f H(C) L (C), then f = const. More generally, if f(z) C(1 + z N ) for all z C, for some fixed integer N and a finite constant C, then f is polynomial of degree at most N. Proof. (a) follows by putting absolute values inside (1.6). For (b) apply (a) to Ω = D(, R) and let R. This shows that f (k) for all k > N. Part (b) has a famous consequence, namely the fundamental theorem of algebra. Proposition 1.2. Every P C[z] of positive degree has a complex zero, in fact it has exactly as many zeros over C (counted with multiplicity) as its degree.

26 26 1. BASIC COMPLEX ANALYSIS I Proof. Suppose P (z) C[z] is a polynomial of positive degree and without zero in C. Then f(z) := 1 P (z) H(C) and since P (z) as z, f is evidently bounded. Hence f = const and so P = const contrary to the assumption of positive degree. So P (z ) = for some z C. Factoring out z z we conclude inductively that P has exactly deg(p ) many complex zeros as desired. Next, we show how Theorem 1.16 allows us to define local primitives. In particular, we can clarify the issue of the logarithm as the local primitive of 1 zṗroposition Let Ω be simply connected. Then for every f H(Ω) so that f everywhere on Ω there exists g H(Ω) with e g(z) = f(z). Thus, for any n 1 there exists f n H(Ω) with (f n (z)) n = f(z) for all z Ω. In particular, if Ω C is simply connected, then there exists g H(Ω) with e g(z) = z everywhere on Ω. Such a g is called a branch of log z. Similarly, there exist holomorphic branches of any n z on Ω, n 1. Proof. If e g = f, then g = f f g(z) := in Ω. So fix any z Ω and define z z f (ζ) f(ζ) dζ where the integration path joins z to z and consists a finite number of line segments (say). We claim that g(z) does not depend on the choice of path. First note that f f H(Ω) due to analyticity. Second, by the simple connectivity of Ω any two curves with coinciding initial and terminal points are homotopic to each other via a piece-wise C 1 homotopy. Thus, Theorem 1.16 yields the desired equality of the integrals. It is now an easy matter to check that g (z) = f (z) f(z). Indeed, g(z + h) g(z) h = 1 f (z + th) f(z + th) dt f (z) f(z) as h So g H(Ω) and (fe g ) shows that e g = cf where c is some constant different from zero and therefore c = e k for some k C. Hence, e g(z) k = f(z) for all z Ω and we are done. Throughout, for any disk D, the punctured disk D denotes D with its center removed. Corollary Let f H(Ω). Then the following are equivalent: f for some z Ω, f (n) (z ) = for all n the set {z Ω f(z) = } has an accumulation point in Ω Assume that f is not constant. Then at every point z Ω there exist a positive integer n and disks D(z, ρ), D(f(z ), r) with the property that every w D(f(z ), r) has precisely n pre-images under f in D(z, ρ). In

27 4. INTEGRATION 27 particular, if f (z ), then f is a local C diffeomorphism. Finally, every nonconstant holomorphic map is an open map (i.e., it takes open sets to open sets). Proof. Let z n z Ω as n, where f(z n ) = for all n 1. let f(z) = a k (z z ) k = a N (z z ) N (1 + O(z z )) as z z k= locally around z where N satisfies a N. But then it is clear that f does not vanish on some disk D(z, r), contrary to assumption. Thus, f locally around z. Since Ω is connected, it then follows that f on Ω. This settles the equivalencies. If f does not vanish identically, let us first assume Figure 1.6. A branch point that f (z ). We claim that locally around z, the map f(z) is a C diffeomorphism from a neighborhood of z onto a neighborhood of f(z ) and, moreover, that the inverse map to f is also holomorphic. Indeed, in view of Theorem 1.4, the differential df is invertible at z. Hence, by the usual inverse function theorem we obtain the statement about diffeomorphisms. Furthermore, since df is conformal locally around z, its inverse is, too and so f 1 is conformal and thus holomorphic. If f (z ) =, the there exists some positive integer n with f (n) (z ). But then f(z) = (z z ) n h(z) with h H(Ω) satisfying h(z ). By Proposition 1.21 we can write h(z) = (g(z)) n for some g H(U) where U is a neighborhood of z and g(z ). Thus, f(z) = ((z z )g(z)) n. Figure 1.6 shows that case of n = 8. The dots symbolize the eight pre-images of some point. Finally, by the preceding analysis of the n = 1 case we conclude that (z z )g(z) is a local diffeomorphism which implies that f has the stated n-to-one mapping property. The openness is now also evident.

28 28 1. BASIC COMPLEX ANALYSIS I We remark that any point z Ω for which n 2 is called a branch point. The branch points are precisely the zeros of f in Ω and therefore form a discrete subset of Ω. The open mapping part of Corollary 1.22 has an important implication known as the maximum principle. Corollary Let f H(Ω). If there exists z Ω with f(z) f(z ) for all z Ω, then f = const. Proof. If f is not constant, then f(ω) is open contradicting that f(z ) f(ω), which is required by f(z) f(z ) on Ω. The maximum principle has numerous important applications as well as variants and extensions. In Chapter 12, we present the simple but powerful Schwarz lemma as an application (see Problem 11), whereas for such extensions as the three lines and circle theorems, as well as the Phragmen-Lindelöf theorems we refer the reader to the classical literature, see [28] and [37], as well as [34] (in fact, a version of the Phragmen-Lindelöf principle is discussed in Chapter 12, see Problem 36). To conclude this chapter, we present Morera s theorem (a kind of converse to Cauchy) and (conjugate) harmonic functions. Especially the latter is one of the central tools of complex analysis and Riemann surfaces. We begin with Morera s theorem. Theorem Let f C(Ω) and suppose T is a collection of triangles in Ω which contains all sufficiently small triangles 3 in Ω. If f(z) dz = T T then f H(Ω). T Proof. The idea is simply to find a local holomorphic primitive of f. Thus, assume D(, r) Ω is a small disk and set F (z) := z f(ζ) dζ = z 1 f(tz) dt for all z < r. Then by our assumption, for z < r and h small, F (z + h) F (z) h = 1 f(z + ht) dt f(z) as h. This shows that F H(D(, r)) and therefore also F = f H(D(, r)). Hence f H(Ω) as desired. Next, we introduce harmonic functions. 3 This means that every point in Ω has a neighborhood in Ω so that all triangles which lie inside that neighborhood belong to T

29 5. HARMONIC FUNCTIONS Harmonic functions Definition A function u : Ω C is called harmonic iff u C 2 (Ω) and u =. Typically, harmonic functions are taken to be real-valued but there is no need to make this restriction in general. The following result explains the ubiquity of harmonic functions in complex analysis. Proposition If f H(Ω) and f = u + iv, then u, v are harmonic in Ω. Proof. First, u, v C (Ω). Second, by the Cauchy Riemann equations, as claimed. u xx + u yy = v yx v xy =, v xx + v yy = u yx + u xy = This motivates the following definition. Definition Let u be harmonic on Ω and real valued. We say that v is the harmonic conjugate of u iff v is harmonic and real valued on Ω and u + iv H(Ω). Let us first note that a harmonic conjugate, if it exists, is unique up to constants: indeed, if not, then we would have a real-valued harmonic function v on Ω so that iv H(Ω). But from the Cauchy Riemann equations we would then conclude that v = or v = const by connectedness of Ω. This definition of course presents us with the question whether every harmonic function on a region of R 2 has a harmonic conjugate function. The classical example for the failure of this is u(z) = log z on C ; the unique harmonic conjugate v with v(1) = would have to be the polar angle which is not harmonic on C. However, in view of Proposition 1.21 it is on every simply connected subdomain of C. As the following proposition explains, this is a general fact. Proposition Let Ω be simply connected and u harmonic on Ω. Then u = Re (f) for some f H(Ω) and f is unique up to an imaginary constant. Proof. We already established the uniqueness property. To obtain existence, we need to solve the Cauchy Riemann system. In other words, we need to find a potential v to the vector field ( u y, u x ) on Ω, i.e., v = ( u y, u x ). If v exists, then it is C 2 (Ω) and hence v harmonic. Define v = u yx + u xy = v(z) := z z u y dx + u x dy